Gudmund Skovbjerg Frandsen

Publications (3)0 Total impact

• Source

  Available from: dagstuhl.de

  Chapter: Dynamic Matrix Rank

  Gudmund Skovbjerg Frandsen, Peter Frands Frandsen
  [show abstract] [hide abstract]
  ABSTRACT: We consider maintaining information about the rank of a matrix under changes of the entries. For n×n matrices, we show an upper bound of O(n 1.575) arithmetic operations and a lower bound of Ω(n) arithmetic operations per change. The upper bound is valid when changing up to O(n 0.575) entries in a single column of the matrix. Both bounds appear to be the first non-trivial bounds for the problem. The upper bound is valid for arbitrary fields, whereas the lower bound is valid for algebraically closed fields. The upper bound uses fast rectangular matrix multiplication, and the lower bound involves further development of an earlier technique for proving lower bounds for dynamic computation of rational functions.
  06/2006: pages 395-406;
• Conference Proceeding: Dynamic Matrix Rank.

  Gudmund Skovbjerg Frandsen, Peter Frands Frandsen
  Automata, Languages and Programming, 33rd International Colloquium, ICALP 2006, Venice, Italy, July 10-14, 2006, Proceedings, Part I; 01/2006
• Source

  Available from: dagstuhl.de

  Article: Dynamic matrix rank

  Gudmund Skovbjerg Frandsen, Peter Frands Frandsen
  [show abstract] [hide abstract]
  ABSTRACT: We consider maintaining information about the rank of a matrix under changes of the entries. For n×n matrices, we show an upper bound of O(n1.575) arithmetic operations and a lower bound of Ω(n) arithmetic operations per element change. The upper bound is valid when changing up to O(n0.575) entries in a single column of the matrix. We also give an algorithm that maintains the rank using O(n2) arithmetic operations per rank one update. These bounds appear to be the first nontrivial bounds for the problem. The upper bounds are valid for arbitrary fields, whereas the lower bound is valid for algebraically closed fields. The upper bound for element updates uses fast rectangular matrix multiplication, and the lower bound involves further development of an earlier technique for proving lower bounds for dynamic computation of rational functions.
  Theoretical Computer Science.